Why so many 'multi-part' definitions, as opposed to 'unified' ones?


$\bullet\;$ As my most recent example, I discovered (through questions here on MSE) that $\;\inf{\cdots}\;$ can simply be defined by postulating $$z \leq \inf{A} \;\equiv\; \langle \forall a : a \in A : z \leq a \rangle$$ for any $\;z\;$ and lower-bounded $\;A\;$. Contrast this with \begin{align} & z \in A \;\then\; \inf{A} \leq z \\ & \langle \forall a : a \in A : z \leq a \rangle \;\then\; z \leq \inf{A} \\ \end{align} or even \begin{align} & z \in A \;\then\; \inf{A} \leq z \\ & \langle \forall \epsilon : \epsilon > 0 : \langle \exists a : a \in A : a < \inf{A} + \epsilon \rangle \rangle \\ \end{align}

$\bullet\;$ For sets, the symmetric difference is often defined as $$A \triangle B \;=\; (A \setminus B) \cup (B \setminus A)$$ or $$A \triangle B \;=\; (A \cup B) \setminus (A \cap B)$$ while in practical proofs I find it much easier to work with $$x \in A \triangle B \;\equiv\; x \in A \;\not\equiv\; x \in B$$ for all $\;x\;$, since $\;\not\equiv\;$ is the logic-level equivalent of $\;\triangle\;$.

$\bullet\;$ The textbook definition of '$\;\mathscr T\text{ is a topology on }X\;$' is that \begin{align} & \mathscr T \subseteq \mathscr P(X) \\ & \emptyset \in \mathscr T \\ & X \in \mathscr T \\ & \mathscr T\text{ is closed under }\cdots \cap \cdots \\ & \mathscr T\text{ is closed under }\bigcup \\ \end{align} However, given closure under $\;\bigcup\;$, the first three conditions can be unified to just $$\bigcup \mathscr T = X$$ which has the very intuitive reading '$\;\mathscr T\;$ covers $\;X\;$'.

$\bullet\;$ In logic, I almost aways see the 'uniqueness quantifier' $\langle \exists! x :: P(x) \rangle$ ('there exists exactly one') defined as $$\langle \exists x :: P(x) \rangle \;\land\; \langle \forall x,y : P(x) \land P(y) : x=y \rangle$$ where $$\langle \exists y :: \langle \forall x :: P(x) \;\equiv\; x = y \rangle \rangle$$ is shorter and often seems much easier to work with. And it has a nice symmetry: the $\;\then\;$ direction of the equivalence is uniqueness, which the $\;\when\;$ direction is existence.

$\bullet\;$ Finally, as an example from various domains, a statement of the form $\;P \equiv Q\;$ is very often seen as an invitation to give separate proofs for $\;P \then Q\;$ and $\;Q \then P\;$; and similarly for mutual inclusion for sets, and for proving equality of numbers using $\;\le\;$ and $\;\ge\;$, or even $\;\lt,=,\gt\;$.

The common pattern in all of the above, is that people seem to prefer 'multi-part' definitions over 'unified' definitions. And I'm wondering why this is.

Does a proof which is split in parts perhaps have a proof-practical advantage? As a kind of counterexample, a while ago I discovered that a relation $\;R\;$ on $\;A\;$ is an equivalence relation exactly when $$aRb \:\equiv\: \langle \forall x :: aRx \equiv bRx\rangle$$ holds for all $\;a,b\;$ (where $\;a,b,x\;$ range over $\;A\;$). However, when I tried to actually use this definition to prove some relation to be an equivalence relation, then almost always the resulting proof was more complex than a proof of the three parts (reflexivity, symmetry, transitivity). So in this specific example, the 'unified' definition did not really help me. But in my experience, this has been the exception: 'unified' definitions almost always really work in practice for me.

Do the parts perhaps have an educational value? Perhaps, at least initially, it is easier to build an intuition using separate parts, and then both those proofs and also later proofs are structured around that 'multi-part' intuition.

Is there perhaps an 'implicational bias'? In other words, is it perhaps that I've been brought up in the 'school' of Dijkstra-Feijen, Gries-Schneider, et al., where there is an emphasis on equality and equivalence and symmetry, while most people approach proofs 'sequentially' based on inferences?

Or is something else at work here?

• None of the other axioms of a topology imply that the empty set is a member of the topology. The case of a one point set illustrates this. – Matt Samuel Feb 15 '16 at 5:42
• @MattSamuel I think they do. Since $\;\mathscr T\;$ is closed under $\;\bigcup\;$, that means that, since $\;\emptyset \subseteq \mathscr T\;$, therefore also $\;\bigcup \emptyset \in \mathscr T\;$, and $\;\bigcup \emptyset = \emptyset\;$. Right? – MarnixKlooster ReinstateMonica Feb 15 '16 at 5:44
• @MattSamuel OK, usually I've seen a topology defined as 'finite [non-empty] intersections and arbitrary [including empty] unions', which directly match closure under $\;\cdots \cap \cdots\;$ and under $\;\bigcup\;$ as I wrote in the question. You say that you usually see the more restricted axiom 'arbitrary but non-empty unions' and then separately an axiom for the empty set? That would emphasize the point of my question even more, I think. – MarnixKlooster ReinstateMonica Feb 15 '16 at 6:10
• Usually it's only implied, not explicit. For most things there's little harm in including unnecessary axioms for clarity, and sometimes achieving maximum economy involves unnecessary obfuscation. – Matt Samuel Feb 15 '16 at 6:13
• @MattSamuel Indeed, Compare those "economic" group axioms where only a left neutral and left inverse is demanded – Hagen von Eitzen Feb 15 '16 at 7:44

2 Answers

Regarding binary relations, there are many important different types. An equivalence relation is symmetric, reflexive, and transitive. A linear order < is anti-symmetric, irreflexive, transitive, and satisfies trichotomy. A well-order is a linear order with an additional condition. A poset (the kind used in the set-theoretic topic called Forcing) is reflexive and transitive. And there are of course many others. Instead of trying to compress the definitions, it is often more useful to list the parts, as it can then be seen how varying the parts results in other structures.

Chess players say "To win you must use all your pieces". When attempting a proof, a list of properties, even if logically redundant, can help you to see some important property that you haven't used.

Sometimes a defining list is easier to use because it incorporates more data: Let $\times$ be an associative binary operation on a set $G\ne \phi$ such that $\forall x,y\in G\;[\;(\exists! z\in G\;(x\times z=y)\land (\exists!z'\in G\; (z'\times x=y) \;].$ It takes some work to show that this meets all the "usual list" of conditions for a group. The usual def'n mentions an identity and unique two-sided inverses.

On the other hand, some writers do present def'ns that are much longer than most people would deem necessary.

I think I have a decent answer. Let's go to the English language for a second. If I just said proopiomelanocortin, do you have any idea what I'm referring to ? but if I define the prefixes and suffixes that make up the word it might be easier to grasp:

$\underline{pro-}:$ to go forward, to lead to.

$\underline{-in}:$a protein ( okay technicality, I don't know if that's it's exact meaning).

$\underline{opio-}:$ like opium.

$\underline{melano-}:$ black,pigmentation.

$\underline{cort-}:$ the outer layer of a body part ( in this case specifically the adrenals)

putting this all together we get:

$\underline {proopiomelanocortin}:$ the protein leading to the production of: opium like substances, pigmentation, and the adrenal cortex hormones.

So, breaking it down, means if you understand the parts, you can get a reasonable grasp on the overall meaning. Building from basics, just allows a broader audience. like in your written definitions above I didn't know $\triangle$ meant what you used it for. So until I learned that, you might as well have been speaking Greek. To understand, what covers means, in topology most times you have to understand, the parts you clumped together, so you would have to define covers to anyone reading it who didn't know what that meant. So in short, it really depends, on who you are aiming for as an audience what definitions make sense to use.